Verifying Temporal Properties of Systems with Applications to
Petri Nets

Julian C Bradfield

Abstract: This thesis provides a powerful general-purpose
proof technique for the verification of systems, whether finite or
infinite. It extends the idea of finite local model-checking, which
was introduced by Stirling and Walker: rather than traversing the
entire state space of a model, as is done for model-checking in the
sense of Emerson, Clarke et al. (checking whether a (finite)
model satisfies a formula), local model-checking asks whether a
particular state satisfies a formula, and only explores the nearby
states far enough to answer that question. The technique used was a
tableau method, constructing a tableau according to the formula and
the local structure of the model. This tableau technique is here
generalized to the infinite case by considering sets of states,
rather than single states; because the logic used, the
propositional modal mu-calculus, separates simple modal and boolean
connectives from powerful fix-point operators (which make the logic
more expressive than many other temporal logics), it is possible to
give a relatively straightforward set of rules for constructing a
tableau. Much of the subtlety is removed from the tableau itself,
and put into a relation on the state space defined by the tableau -
the success of the tableau then depends on the well-foundedness of
this relation.

This development occupies the second and third chapters: the
second considers the modal mu-calculus, and explains its power,
while the third develops the tableau technique itself.

The generalized tableau technique is exhibited on Petri nets,
and various standard notions from net theory are shown to play a
part in the use of the technique on nets - in particular, the
invariant calculus has a major role.

The requirement for a finite presentation of tableau for
infinite systems raises the question of the expressive power of the
mu-calculus. This is studied in some detial, and it is shown that
on reasonably powerful models of computation, such as Petri nets,
the mu-calculus can express properties that are not merely
undecidable, but not even arithmetical.

The concluding chapter discusses some of the many questions
still to be answered, such as the incorporation of formal reasoning
within the tableau system, and the power required of such
reasoning.